Oxbridge vs LSE

I’m a rising senior at Yale with a strong GPA 3.92/4, high GRE, and very good extracurriculars. I was just accepted in to Harvard Law School’s junior deferral program, so I would like to pursue an MPP in the mandatory gap period my undergraduate and law. Between Oxford, Cambridge, and LSE which is best for transatlantic relations. Further, beyond a research assistant position and summer think tank work, I have no work experience. How much will this hurt my chances?